Wavelet Inpainting Based on Tensor Diffusion

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1 Vol. 39, No. 7 ACTA AUTOMATICA SINICA July, 03 Wavelet Inpainting Based on Tensor Diffusion YANG Xiu-Hong GUO Bao-Long WU Xian-Xiang Abstract Due to the lossy transmission in the JPEG000 image compression standard, the loss of wavelet coefficients heavily affects the quality of the received image. In this paper, we propose a novel wavelet inpainting model based on tensor diffusion TDWI) to restore the missing or damaged wavelet coefficients. A hybrid model is built by combining structure-adaptive anisotropic regularization with wavelet representation. Its associated Euler-Lagrange equation is also given for analyzing its regularity performance. Owing to the matrix representation of the structure tensor in the regularization term, the shape of diffusion kernel changes adaptively according to the image features, including sharp edges, corners and homogeneous regions. Compared with existing wavelet inpainting models, the proposed one can control more adaptively and accurately the geometric regularity in the image and exhibits better robustness to noise. In addition, an effective and proper numerical scheme is adopted to improve the computation. Experimental results on a variety of loss scenarios are given to demonstrate the advantages of our proposed model. Key words Wavelet, inpainting, structure tensor, diffusion, anisotropic, regularization Citation Yang Xiu-Hong, Guo Bao-Long, Wu Xian-Xiang. Wavelet inpainting based on tensor diffusion. Acta Automatica Sinica, 03, 397): DOI 0.374/SP.J Image inpainting refers to as filling in missing or damaged information in an image. It plays an important role in computer vision and image processing, including image replacement [], disocclusion [ 3], and error concealment [4 6]. The missing information can occur in the pixel domain, as well as the transformed domain. After the release of the image compression standard JPEG000, wavelet transform plays an important role in image formatting, storing, compression and transmission coding. During the lossy storing or transmission, the loss of wavelet packets in received subbands may heavily affect the image quality. Therefore, it is necessary to recover the source image by using the incomplete wavelet data, which can be addressed as wavelet inpainting originally proposed in [7], since it is closely related to the classic image domain inpainting. However, wavelet inpainting differes from its image domain counterpart. For image domain inpainting, the clear cut inpainting regions are easy to be localized and various kinds of diffusion schemes based on variational partial differential equation PDE) models can be directly used to restore the damaged regions. For wavelet inpainting, however, the loss of wavelet coefficients in some subband affects the whole image quality. Therefore, the resulting damaged regions cannot be localized simply in the pixel domain. This challenge prohibits the direct application of the existing PDE models [8 9] to wavelet inpainting. Besides, the property of wavelet decoupling makes it insufficient to directly interpolate in the wavelet domain. Chan et al. [7] proposed two related variational models for wavelet inpainting, which combine the total variation TV) minimization technique with wavelet representation. Their main idea is to use the TV regularization in the pixel domain to control and restore the wavelet coefficients in the wavelet domain. The main benefit of the TV regularization is that it can preserve edges very well [8], but it suffers from the staircase artifact. Aiming to overcome the defect, hang et al. [0] adopted the p-laplace operator which gives better diffusion performance. Yau et al. [] took the L 0-norm in the wavelet domain and then filled the missing information by Manuscript received September 5, 0; accepted January 9, 03 Supported by National Natural Science Foundation of China , ) and the Fundamental Research Funds for the Central Universities K ) Recommended by Associate Editor LIU Yi-Jun. ICIE Institute, School of Electromechanical Engineering, Xidian University, Xi an 7007, China the TV minimization. Hadji et al. [] used a tixotrop model to make use of the smoothing model and correct the lying out of the contours by putting them more clearly. hang et al. [3] used a nonlocal TV operator to extend the Chan, Shen and hou s TV wavelet inpainting models. Although existing regularization methods are quite efficient for recovering damaged wavelet coefficients as presented in [7, 0 3], it is well known that these regularization terms only use local gradient magnitude as a measure to control diffusion performance without using the direction information of the structures in the image. In this paper, we explore the feasibility of structure-adaptive anisotropic regularization to exploit both structure magnitude and direction information in wavelet inpainting. The anisotropic structure tensor uses the matrix representation of the gradient, thus we can accurately estimate the strength of structures and the direction of the maximum variation in the image [4 8]. Accordingly, we can design a matrix-valued diffusion tensor to control the amount of diffusion and the orientation of smoothing. Therefore, instead of using local gradient based regularizations, we use the structureadaptive anisotropic regularization to control the geometric regularity in the pixel domain to restore the damaged wavelet coefficients. Our main contribution in this paper is to present a wavelet inpainting model based on tensor diffusion and build up its associated Euler-Lagrange equation to analyze its structure-adaptive anisotropic regularization. The proposed model is solved by an efficient scheme. The advantages of this effective model for different loss cases are demonstrated by a large number of numerical simulations. This paper is organized as follows. In Section, we propose the TDWI model. In Section, according to its associated Euler-Lagrange equation, we compare the performance of the TDWI model with other wavelet inpainting models. Section 3 gives the implementation scheme for the proposed model. Generic numerical examples in Section 4 further highlight the remarkable inpainting qualities of the proposed model. The Appendix A gives the derivation of the Euler-Lagrange equation of the proposed model. Wavelet inpainting model based on tensor diffusion A standard image model is given by u x) = f x) + n x) )

2 07 ACTA AUTOMATICA SINICA Vol. 39 where x = [x, x ] T = [, M] [, N], f x) is the original intact image and n x) is the additive Gaussian white noise. Let us denote the standard wavelet representations of f x) and u x) by f α, x) = and α ψ x), α = {α, j, k u β, x) = β ψ x), β = {β, j, k 3) where {ψ denotes the wavelet basis function For tensorproduct based multiresolution analysis in D, there are three directions of wavelet functions ψ ), ψ ), and ψ 3), but for convenience a single symbol ψ is used instead). α = {α and β = {β are the wavelet coefficients of f x) and u x), respectively. The coefficients α are the values being stored and transmitted in wavelet based image representation and transmission. Loss in different wavelet subbands in α on the index region I will cause spatially inhomogeneous degradation. The task of wavelet inpainting is to recover the missing or damaged wavelet coefficients {α on ) I in an appropriate manner [9 ]. We present a new wavelet inpainting model based on tensor diffusion as follows: minf u, f) = tr Ψ J)) + λ β α ) { λ, ) \I λ = 0, ) I 4) where parameter λ is a positive constant and tr ) is the trace operator sum of the diagonal elements). In 4), J is the structure tensor which is defined as [ ] [ J = u u T u u = ux) ] u = xu y u u u xu y u y) 5) where the gradient u = u x,u y) T. Since J is easily polluted by noise, we could use proper filtering techniques to smooth the tensor data. J is a symmetric positive semidefinite matrix. Hence, there exist two nonnegative eigenvalues µ and µ setting µ µ ) and the corresponding mutually orthogonal eigenvectors v and v. Thus, J can also be expressed as J = µ v v T + µ v v T Its eigenvalues specify the variations of the image in the directions of their corresponding eigenvectors. The eigenvector v is orthogonally aligned to the edges of an image, while the eigenvector v determines the dominant orientation of the local structure. Therefore, J can give accurate local analysis of image structures [4, 7]. In 4), Ψ s ) is a penaliser which is a differentiable and increasing function that is convex in s and its derivative satisfies Ψ s ) = g s ), so we choose it as Ψ s ) = εs + ε) γ + s 7) γ Correspondingly, g s ) = ε + ε) ) 6) + s γ 8) where 0 < ε <, γ > 0, thus, function g s ) must be a decreasing function. Both Ψ s ) and g s ) are scalar-valued functions, and then we generalize them to matrix-valued functions as follows: Ψ J) = T Ψ µ i) v iv i 9) i g J) = i g µ i) v iv i T Analysis of the proposed model 0) For the energy functional F u, f) in 4), the corresponding Euler-Lagrange equation which is deduced in Appendix A) is given by g J) u β, x)), ψ + λ β α ) = 0 ) where denotes the divergence operator, and, denotes the inner product. The corresponding gradient descent flow of ) with an artificial time variable τ is β ) τ = g J) u β, x)), ψ λ β α ) ) The first term of ) combines structure-adaptive anisotropic regularization with wavelet decomposition. The regularity in the pixel domain is dependent on the term below: tu = g J) u β, x)) = D u β, x)) 3) where D = g J) is referred to as a diffusion tensor. According to 0), we can obtain that D = g J) = i g µ i) v iv i T = i k iv iv i T 4) where k = g µ ) and k = g µ ). Therefore, k, v ) and k, v ) are the eigenpairs of the diffusion tensor D. As in Fig. a), v = cos θ, sin θ) T and v = sin θ, cos θ) T, where θ is the direction of v. Fig. The Gaussian kernels Since the diffusion process with diffusion time τ equals the Gaussian convolution with standard deviation ρ = τ, we can get the corresponding Gaussian convolution kernel of 3) for pixel x as u x, τ) = u x, 0) G ρ ρ= τ G ρ ρ= τ = 4πτ exp ) x T D x 5) 4τ

3 No. 7 YANG Xiu-Hong et al.: Wavelet Inpainting Based on Tensor Diffusion 073 where denotes the convolution operator. According to 4), we know that D is also a symmetric and positive semidefinite matrix, then we obtain D = v iv T i = k i i cos θ + sin θ sinθcosθ ) k k k k sinθcosθ ) sin θ + cos θ k k k k 6) ) ) where k, v and k, v are the eigenpairs of D. Thus, ) ) x T D cos θ x = + sin θ x sin θ + + cos θ x + k k k k sinθcosθ k k )) x x 7) Since D is a real symmetric matrix, there exists the orthogonal transformation of 7). Therefore, we can get its quadratic form according to its normalized form that x T D x = y k + y k 8) Thus, the Gaussian kernel G ρ is y G ρ ρ= τ = 4πτ exp + y k k 4τ 9) As shown in Fig., we can analyze the shape of the Gaussian kernel according to the local structure feature in the image: ) When x belongs to the regions with a linear structure, we can obtain that µ µ 0 and k k. Thus the shape of the Gaussian kernel is a highly oriented ellipse whose main axis is parallel to the dominant orientation v, which leads to structure-adaptive anisotropic regularization and allows to smooth along image edges while inhibiting smoothing across edges. ) When x belongs to the homogeneous regions, we can obtain that µ µ 0 and k k 0. Thus the shape of the Gaussian kernel is a large circular, which leads to a strong isotropic regularization. 3) When x belongs to the regions with a corner or junction, we can obtain that µ µ 0 and k k 0. Thus the kernel s shape is a small circular, which leads to a weak isotropic regularization. Therefore, the regularization property depends on the Gaussian kernel whose shape changes adaptively according to the local structure feature. Besides, the integration of local orientation makes it applicable to distinguishing areas with edges from areas with corners. Hence, the proposed model can even perform adaptive regularization at the presence of corners. Next, we analyze the physical characteristics of other anisotropic diffusion operators which also consider the structure information, such as TV regularization in [7 8], the p-laplace regularization in [0], nonlocal TV regularization in [3] and the p CDD operator in [9]. TVu = u u ) = u u ξξ 0) pu = u p u ) = u p u ξξ + p ) u p u ηη ) NLTVu = w wu wu ) ) where ξ = u/ u and η = u/ u. In the local orthogonal coordinates ξ, η), as shown in Fig., the η-axis is parallel to the gradient direction and the ξ-axis is perpendicular. Fig. The local coordinates ξ, η) vs. the whole coordinates x, x ) Since TV operator TVu only diffuses along ξ with diffusion coefficient u, it can keep edges very well, but it suffers from the staircase effect. The diffusion performance of p-laplace operator pu < p < ) in the directions of ξ and η are determined by u p and p ) u p, respectively. Due to the constant parameter p, its diffusion amounts in the two directions are both small at the presence of image structures and both large in homogeneous regions. Nonlocal TV regularization NLTVu is the extension of TVu in terms that NLTVu considers a more general neighbor system here w denotes the weighted graph operator), thus both of them have similar diffusion performance. In [9], the p-laplace operator pu is used to replace the curvature in the curvature-driven diffusion CDD) model to restore the damaged image in the pixel domain. The curvature κ = u/ u ) is the limitation of pu when p, thus the PDE model in [9] is formulated as ) pu p CDDu = u u 3) Although the geometric information is additionally taken into account to decide the diffusion strength, the diffusivity pu / u is also scalar-valued as the same as 0) ). It does not utilize the direction information of the structure to control the diffusion process either. Compared with the four regularizations in 0) 3), the structure-adaptive anisotropic regularization adopted in the proposed model uses the matrix-valued diffusion tensor to exploit more structure information. It can adaptively control the geometric regularity in the pixel domain according to both structure strength and structure direction. The matrix representation of the structure tensor allows the integration of information from a local neighborhood, which leads to better robustness to noise. Thus, its estimations of the diffusion orientation and diffusion amount are more accurate and robust. 3 Implementation scheme Instead of the unconstrained model 4), we consider its constrained version as { β = arg min tr Ψ J)) u s.t. P Ī β) α ε 4)

4 074 ACTA AUTOMATICA SINICA Vol. 39 { β, ) \I where P Ī β) =. 0, else Let A = P Ī W, where W denotes the forward wavelet transform. By Bregman iteration, problem 4) can be solved by the following iteration scheme: { u n+ = arg min µ tr Ψ J)) + u Au βn β n+ = β n + β Au n+) 5) for a positive parameter µ > 0. Here, β n = β for noise-free case. The first subproblem is solved by the proximal forwardbackward splitting PFBS) algorithm as used in [3]. The idea of PFBS is to solve a sum of two convex functionals by one-step forward gradient descent on one functional and one-step backward inverting on another functional. Let u n+,0 = u n and for l 0, { u n+,l+ = arg min u tr Ψ J)) + u u n+,l ηa T Au n+,l β n)) µη 6) where η is a positive parameter and satisfies 0 < η < / A T A. A T = W T P T Ī, where P T Ī is the zero-padding operator and W T can be implemented by the inverse wavelet transform W. Finally, problem 4) can be solved by the two-step iteration scheme as follows: v n+,l+ = u n+,l ηw P T Ī P Ī W u n+,l β n) 7a) { u n+,l+ = arg min tr Ψ J)) + u u v n+,l+ 7b) µη 7b) can be solved by the gradient descent flow in the pixel domain. Let V = v n+,l+ ; its corresponding Euler- Lagrange equation is g J) u) + u V ) = 0 8) µη Let U = u n+,l+ ; 8) can be solved by the following non-linear PDE: U = g J) u) t µη u V ) 9) u x,0) = u n+,l We use the following PDE to obtain the smoothed tensor data J as in [6], u i,j x,τ) = g R σ) u i,j), i, j =, t 30) R σ = G σ J where the diffusivity matrix R σ is obtained by the convolution of each channel of the initial tensor with the Gaussian kernel G σ. We use a simple explicit scheme to solve the PDE and the time step size is denoted by t. Algorithm. { ) Start with n = 0, initial guess β new = α χ, χ = 0, j, k) I, j, k) / I, βnew = ) β new, β 0 = β new; Set β old = 0, β old = ) β old ; E = βnew β old. ) While n N outer and not stopping criterion, do a) β old = β new, set u 0 = W β old ). b) J 0 = u 0) [ ] u u 0)T 0 = u 0 u 0 u 0 and R σ = G σ J 0. c) for m = : N, do i) Ppq m = g ) R σ) u m pq, p, q =, which satisfies the boundary condition: ) g R σ) u m pq n = 0 on, where n is the normal vector on. ii) u m pq = [ u m pq + tp ] pq m, p, q =,. u m iii) J = u m u m u m. end the for loop. d) for l = 0 : N inner, do i) u n+,0 = u n. ii) v n+,l+ =u n+,l ηw P ) T Ī P Ī W u n+,l β old. iii) Let V = v n+,l+, solving u n+,l+ using 9) u n+,l+ = arg min u { tr Ψ J)) + u V µη end the for loop. e) new image u n+ = u n+,n inner. f) update β new = β old + β 0 P Ī W u n+, n = n +. end the while loop. For noise-free case, the iteration scheme should be taken as β old = β 0 in d)ii) step above and the stopping criterion is PĪ W u β For noisy case, the stopping criterion is E 0. 4 Experimental results In this section, we simulate the proposed algorithm on Matlab R009b. In all simulations, we use Daubechies 7/9 biorthogonal wavelet with symmetric extension, which is used in the standard JPEG000 for lossy compression and transmission. The standard peak signal to noise ratio PSNR) as shown in 3) is employed to objectively evaluate the proposed model: PSNR = 0lg 55 u u 0 ) db 3) where u 0 and u are the original image and the inpainted image, respectively. We compare the proposed algorithm TDWI) with existing wavelet inpainting algorithms in [7, 0, 3] abbreviated as TVWI, p-laplacewi, and NLTVWI, respectively). t, N, and σ in TDWI are used to obtain the diffusion matrix. These parameters have a limited impact on the wavelet inpainting results. We use t = 0.0, N = 3 and σ = 0. since empirically these choices give good results. According to [3], we set µ = 0.05, λ = 0. for TVWI, and µ = 0.0 for NLTVWI. By default, parameter η used in proximal iteration is set to η =. The wavelet decomposition level is 3.

5 No YANG Xiu-Hong et al.: Wavelet Inpainting Based on Tensor Diffusion 075 The comparison of wavelet inpainting WI) results We compare the WI results with the same iteration step for all the four WI models in various loss cases. We set N outer = 50 and N inner = 0 for all the models. ) Block loss Fig. 3 shows the boat image with block loss and its restored results. In JPEG000, the image is transformed into wavelet subbands and these wavelet coefficients are divided into codeblocks. During the transmission process, the loss or corruption of some codeblocks could affect the whole image quality. Fig. 3 a) is the intact original image; Fig. 3 b) is the block loss mask in the wavelet domain, black squares indicate each level of wavelet decomposition with top-left corner storing the coarsest low-low subband, which is referred to as LL, and the bottom-right storing the corresponding high-high ones. Fig. 3 c) is the received image with the block loss. We can see that the loss in the LL subband causes the severest degradation and results in the black square stains in the image. The loss of other high frequencies in the coarsest subband creates Gibbs artifacts or other blur effects, while the loss of the rest frequency subbands do not greatly affect the image quality. From Figs. 3 d) 3 g), all these wavelet inpainting methods can fill in the missing information properly, but there is some residual blurry ghosting in black square regions in the TVWI, p-laplacewi and NLTVWI reconstructed images, while TDWI can reconstruct perfectly the edges and corners and obtain better visual quality. For the zoomed sub-images, there emerge staircase artifacts in the flat region for TVWI, there exists blurring effect for p-laplacewi, and there is an incompatible region in the black square. The proposed TDWI achieves a higher PSNR and a better visual quality than the other three methods. ) Random loss Fig. 4 shows the Mickey image with random loss and its inpainted images. In Fig. 4 b), black squares indicate each level of wavelet decomposition ditto). Because of the loss in LL frequency, the received image is of quite poor quality and irregular chunky stains emerge in Fig. 4 c). From the subjective respect, TDWI restores the missing homogeneous information more properly and recovers better image structures simultaneously, including edges and corners. From the objective quantification respect, TDWI has an even higher PSNR than TVWI, p-laplacewi and NLTVWI. In Fig. 5, we plot PSNR vs. the retained ratio for the case of random loss in the whole wavelet domain on the Mickey image. It shows that all the models can substantially improve image qualities. However, TDWI outperforms the other three and gives a higher PSNR at ratios from 0 % to 90 %. The proposed model can control the geometric regularity better in the pixel domain. 3) Noisy case To further compare the models, we test the case when there is Gaussian noise in the random loss data. In Fig. 6, we introduce the Gaussian noise with standard deviation to a synthetic image. We can see that compared with the other three, TDWI recovers better not only the inner area but also the edges and corners. Besides, it gives a better denoising result due to its robustness to noise. In Fig. 7, we plot PSNR vs. the retained coefficient ratio for the case of random loss with noise on the synthetic image. TDWI outperforms the other three and gives a higher PSNR at ratios from 0 % to 90 %, since the regularization in the proposed model has better robustness to noise. Fig. 3 The boat image with block coefficients lost a) Original image 5 5; b) Block loss mask; c) Received damaged image PSNR = 6.098); d) Restored image by TVWI PSNR = ); e) Restored image by p-laplacewi PSNR = 7.78); f) Restored image by NLTVWI PSNR = 9.943); g) Restored image by TDWI PSNR = ); h) k) are the zoomed sub-images for d) g), respectively.)

6 076 ACTA AUTOMATICA SINICA Vol. 39 Fig. 5 PSNR vs. the retained ratio on Mickey image without noise) TDWI performs more effectively than the other three and gives a higher PSNR at each retained ratio.) 4. Comparison of execution time We compare the execution time for all the four WI models with the same number of iteration steps. Table shows the execution time with N outer = 50 and N inner = 0 in the cases of Figs. 3, 4, and 6. Table Execution time Time s) TVWI p-laplacewi NLTVWI TDWI Boat Mickey synthetic Since the computation of the diffusion matrix is timeconsuming, the execution time of TDWI is slightly longer than the ones of TVWI and p-laplacewi. However, since the computation of the nonlocal weights in NLTVWI is very expensive, TDWI is much faster than NLTVWI. In addition, we compare the convergence speeds of the four models in the random loss case of Fig. 4. We set a fixed number of the inner iteration steps as N inner = 0. Since the outer iteration performs the diffusion process, we test the PSNR of each WI model with N outer changing. Fig. 8 shows the relationship between PSNR and N outer. In the figure, TDWI converges to a stable state with much fewer iteration steps than the other three models. The tensor diffusion exploits more structure information to perform the structure-adaptive anisotropic regularization in the pixel domain. The proposed TDWI model adjusts adaptively the diffusion amount and diffusion direction according to the structure feature. Therefore, TDWI can give a better convergence speed. Besides, Fig. 8 shows that all the WI models reach the steady state when N outer = 50. Therefore, it is reasonable and fair that we compare the WI results by setting N outer = 50 and N inner = 0 for all the four models in Subsection 4.. Fig. 4 The Mickey image with 50 % wavelet coefficients randomly lost a) Original image 56 56; b) Random loss mask; c) Received damaged image PSNR = 9.493); d) Restored image by TVWI PSNR = ); e) Restored image by p-laplacewi PSNR = 0.98); f) Restored image by NLTVWI PSNR =.5846); g) Restored image by TDWI PSNR = 4.445); h) k) are the zoomed sub-images for d) g), respectively.) 5 Conclusion We proposed a new and effective wavelet inpainting model based on tensor diffusion which combines the structure-adaptive anisotropic regularization with wavelet representation. According to its corresponding EulerLagrange equation, we analyzed the performance of the newly adopted regularization in local coordinates. We discussed different loss scenarios on different types of images,

7 No. 7 YANG Xiu-Hong et al.: Wavelet Inpainting Based on Tensor Diffusion 077 with or without noise. The numerical examples have shown that the proposed model is very effective in filling in missing wavelet information and removing noise. The subjective quality and objective evaluation of the inpainted results are both improved greatly. Since the tensor diffusion exploits more structure information, the proposed model can reach a steady state with much fewer iteration steps. Fig. 8 PSNR vs. the outer iteration step Appendix A Deducing the Euler-Lagrange equation of the proposed model Let βi = {β ) I A) Some energy functional is of the type X λ x+ F u βi, x )) = tr Ψ J)) β α ) Ω A) It follows that Fig. 6 The synthetic image with noise standard deviation $ = 0.5) and 50 % coefficients random loss a) Original image 56 56); b) Received damaged image PSNR=.948); c) Restored image by TVWI PSNR=8.88); d) Restored image by p-laplacewi PSNR=0.650); e) Restored image by NLTVWI PSNR=.835); f) Restored image by TDWI PSNR=3.8608); g) j) are the zoomed sub-images for c) f), respectively.) Fig. 7 PSNR vs. retained coefficient ratio on synthetic image, noise level $ = 0.5) F u βi, x )) = β X J x+ λ β α ) tr Ψ0 J) Ω β A3) Its first term is J x= tr Ψ0 J) β Ω ³ T u β, x ) u β, x )) x= tr Ψ0 J) Ω β u β, x ) tr Ψ0 J) u β, x ))T + Ω β u β, x ))T x= u β, x ) β u β, x ) tr Ψ0 J) u β, x ))T + Ω β T # µ u β, x ) x= u β, x ) β tr Ψ0 J) Ω

8 078 ACTA AUTOMATICA SINICA Vol. 39 [ ψ u β, x)) T + u β, x) ψ ) T] References A4) We know that Ψ J) = l Ψ µ l ) v l v T l, therefore, tr Ψ J) ) J = β tr Ψ µ l ) v l v T l l [ ψ u β, x)) T + u β, x) ψ ) T] = tr ) Ψ µ l ) [v T l ψ v l u β, x)) T) + l ) v T l u β, x) v l ψ ) T)] A5) For column vectors a and b, tr ab T) = a T b and tr a T) = tr a), therefore, tr Ψ J) ) J = β ) )] Ψ µ l ) [ u β, x)) T v l v T l ψ = l u β, x)) T l Ψ µ l ) v l v T l u β, x)) T Ψ J) ψ = u β, x)) T g J) ψ ) ψ = A6) Since g J) is a symmetric matrix, we can obtain that tr Ψ J) ) J = g J) u β, x)) T ψ β A7) We assume that the mother wavelet here we use Daubechies 7 9) is compactly supported and at least Lipschitz continuous, therefore we can yield through integration-by-parts that tr Ψ J) ) J = β g J) u β, x)) T ψ For a vector a, a T) = a), thus tr Ψ J) ) J = β g J) u β, x)) ψ = g J) u β, x)), ψ A8) A9) The Euler-Lagrange equation of the proposed model is Xiong X S, hao H, Tang C J. Image inpainting used in specific target replacement. In: Proceedings of the 008 IEEE International Conference on Information and Automation. Changsha, China: IEEE, Tauber, Li N, Drew M S. Review and preview: disocclusion by inpainting for image-based rendering. IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews, 007, 374): Wighton P, Lee T K, Atkins M S. Dermascopic hair disocclusion using inpainting. In: Proceedings of the 008 International Society for Optics and Photonics. San Diego, CA, USA: SPIE, Ma M Y, Au O C, Chan S H G, Sun M T. Edge-directed error concealment. IEEE Transactions on Circuits and Systems for Video Technology, 00, 03): Lakshman H, Ndjiki-Nya P, Koppel M, Doshkov D, Wiegand T. An automatic structure-aware image extrapolation applied to error concealment. In: Proceedings of the 6th IEEE International Conference on Image Processing. Cairo, Egypt: IEEE, Wu G L, Chen C Y, Chien S Y. Algorithm and architecture design of image inpainting engine for video error concealment applications. IEEE Transactions on Circuits and Systems for Video Technology, 0, 6): Chan T F, Shen J H, hou H M. Total variation wavelet inpainting. Journal of Mathematical Imaging and Vision, 006, 5): Sakurai M, Kiriyama S, Goto T, Hirano S. Fast algorithm for total variation minimization. In: Proceedings of the 8th IEEE International Conference on Image Processing. Brussels, Belgium: IEEE, Li S L, Wang H Q. Image inpainting using curvature-driven diffusions based on p-laplace operator. In: Proceedings of the 4th International Conference on Innovative Computing, Information and Control. Kaohsiung, Taiwan, China: IEEE, hang H Y, Peng Q C, Wu Y D. Wavelet inpainting based on p-laplace operator. Acta Automatica Sinica, 007, 335): Yau A C, Tai X C, Ng M K. L 0-norm and total variation for wavelet inpainting. In: Proceedings of the nd International Conference on Scale Space and Variational Methods in Computer Vision. Voss, Norway: Springer, Hadji M L, Maouni M, Nouri F. A wavelet inpainting by a Tixotrop model. In: Proceedings of the 4th International conference on information visualization. London, United Kingdom: IEEE, hang X Q, Chan T F. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 00, 4): 9 0 g J) u β, x)), ψ + λ β α ) = 0 A0) 4 Brox T, Weickert J, Burgeth B, Mrázek P. Nonlinear structure tensors. Image and Vision Computing, 006, 4): 4 55

9 No. 7 YANG Xiu-Hong et al.: Wavelet Inpainting Based on Tensor Diffusion 5 Yoo H, Kim B, Sohn K. Coherence enhancing diffusion filtering based on connected component analysis structure tensor. In: Proceedings of the 6th IEEE Conference on Industrial Electronics and Applications. Beijing, China: IEEE, Hahn J, Lee C O. A nonlinear structure tensor with the diffusivity matrix composed of the image gradient. Journal of Mathematical Imaging and Vision, 009, 34): Wang Y K, Lu B B, Miao C L. Image fusion based on nonlinear structure tensor. In: Proceedings of the 0 International Conference on Multimedia Technology. Hangzhou, China: IEEE, Gao C Q, Tian J W, Wang P. Generalised-structure-tensorbased infrared small target detection. Electronics Letters, 008, 443): Rane S D, Sapiro G, Bertalmio M. Structure and texture filling-in of missing image blocks in wireless transmission and compression applications. IEEE Transactions on Image Processing, 003, 3): Chan R H, Wen Y W, Yip A M. A fast optimization transfer algorithm for image inpainting in wavelet domains. IEEE Transactions on Image Processing, 009, 87): hang Wen-Juan, Feng Xiang-Chu, Wang Xu-Dong. Mumford-Shah model based on weighted total generalized variation. Acta Automatica Sinica, 0, 38): 93 9 in Chinese) 079 YANG Xiu-Hong Ph. D. candidate at the School of Electromechanical Engineering, Xidian University. She received her bachelor degree and master degree from Xi0 an University of Architecture and Technology in 006 and 009, respectively. Her research interest covers digital image processing and multimedia communication, and digital image restoration. Corresponding author of this paper. yangxiuhong084@yahoo.com.cn GUO Bao-Long Professor at the School of Electromechanical Engineering, Xidian University. He received his bachelor, master and Ph. D. degrees from Xidian University in 984, 988 and 995, respectively. His research interest covers intelligent control and image engineering. blguo@xidian.edu.cn WU Xian-Xiang Ph. D. and associate professor at the School of Electromechanical Engineering, Xidian University. His research interest covers intelligent information processing, and computer vision. wuxianxiang@63.com